Bias-Flip Technique for Frequency Tuning of PiezoElectric Energy Harvesting Devices
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Citation
Zhao, Jianying, Yogesh Ramadass, Jeffrey Lang, Jianguo Ma,
and Dennis Buss. “Bias-Flip Technique for Frequency Tuning of
Piezo-Electric Energy Harvesting Devices.” Journal of Low
Power Electronics and Applications 3, no. 2 (June 2013):
194–214.
As Published
http://dx.doi.org/10.3390/jlpea3020194
Publisher
MDPI AG
Version
Final published version
Accessed
Wed May 25 22:15:26 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/90922
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J. Low Power Electron. Appl. 2013, 3, 194-214; doi:10.3390/jlpea3020194
OPEN ACCESS
Journal of
Low Power Electronics
and Applications
ISSN 2079-9268
www.mdpi.com/journal/jlpea
Article
Bias-Flip Technique for Frequency Tuning of Piezo-Electric
Energy Harvesting Devices
Jianying Zhao 1, Yogesh Ramadass 2, Jeffrey Lang 3, Jianguo Ma 1 and Dennis Buss 2,3,*
1
2
3
Tianjin University, School of Electronic Information Engineering, 92 Weijin Rd, Nankai Dist.,
Tianjin, China; E-Mails: jyzhao@tju.edu.cn (J.Z.); majg@tju.edu.cn (J.M.)
Texas Instruments, Inc., 12500 TI Blvd., Dallas, TX 75243, USA; E-Mail: Yogesh.Ramadass@ti.com
Massachusetts Institute of Technology, EECS Dept, 77 Mass Ave, Cambridge, MA 02139, USA;
E-Mail: lang@mit.edu
* Author to whom correspondence should be addressed; E-Mail: buss@ti.com;
Tel.: +1-617-253-0956.
Received: 13 March 2013; in revised form: 7 May 2013 / Accepted: 24 May 2013 /
Published: 18 June 2013
Abstract: Devices that harvest electrical energy from mechanical vibrations have the
problem that the frequency of the source vibration is often not matched to the resonant
frequency of the energy harvesting device. Manufacturing tolerances make it difficult to
match the Energy Harvesting Device (EHD) resonant frequency to the source vibration
frequency, and the source vibration frequency may vary with time. Previous work has
recognized that it is possible to tune the resonant frequency of an EHD using a tunable,
reactive impedance at the output of the device. The present paper develops the theory of
electrical tuning, and proposes the Bias-Flip (BF) technique, to implement this tunable,
reactive impedance.
Keywords: energy harvesting; bias-flip; piezo-electric
1. Introduction
Figure 1 shows a schematic of a Piezoelectric (PZ) Energy Harvesting Device (EHD) that is the
subject of this research. This structure is referred to as a cantilever structure, and is used to amplify the
J. Low Power Electron. Appl. 2013, 3
195
amplitude of the source vibration [1]. Previous work has shown that maximum output power is
achieved when the cantilever has a high Q resonance at the frequency of the source vibration.
However, the frequency of the source vibration is not usually matched to the resonant frequency of the
EHD. The source vibration may vary with time. This paper addresses the problem of electronically
tuning the PZ EHD to achieve maximum power in situations where the source vibration is not a stable
frequency matched to the mechanical resonant frequency of the EHD.
Figure 1. Schematic of a Piezo-Electric (PZ) Energy Harvesting Device (EHD) based on
the Cantilever Beam structure.
Strain
Strain
+
–
Voltage
In the interest of simplicity, we will analyze the structure in Figure 2. The results achieved through
analysis of this structure can be generalized to the cantilever structure through the addition of
geometrical constants.
Figure 2. Schematic of the simplified EHD that is analyzed in this paper. Ap is the area of
the PZ capacitor, and tp is the thickness. Z is the complex amplitude of the source
vibration, and X is the complex amplitude of the mechanical displacement of the mass M.
This simplified model illustrates the concepts of electronic tuning that apply to the
cantilever structure of Figure 1.
Ap
i
M
X
+
E, D
, 
V
L
RL
tp
‐
Z
This paper describes three concepts for electrically tuning of PZ EHDs.
1. Use of voltage amplitude to tune the mechanical stiffness of the EHD;
2. Coupling of the mechanical resonator to an electrical RLC tank circuit;
3. Bias-Flip (BF) technique to emulate the large tunable inductor that is required for the RLC
tank circuit.
J. Low Power Electron. Appl. 2013, 3
196
These three concepts were introduced in summary form in [2]. In the succeeding sections of this
paper, these concepts will be presented in more detail. Section 6 shows that BF can be used to
effectively optimize the power output from a PZ EHD. In this paper, we have changed some of the
notation that we used in [2], in order to conform to generally accepted usage.
In this paper, we will analyze the PZ EHD. However, many of the results and conclusions are
equally applicable to electromagnetic and electrostatic EHDs. Cammarano et al. [3] have described
concepts very similar to #1 and #2 above in the context of electro-magnetic EHDs.
2. Frequency Tuning by Voltage
The material equations for PZ material can be written as follows [1].


 dE
(1)
D  E  d
(2)
Y
The parameters are defined below.
  mechanical strain (displacement/length)
  mechanical stress (force/area)
Y  Young’s Modulus (force/area)
d  piezo-electric (PZ) coefficient (m/volt)
Y
2  d2

2

 PZ coupling constant
1 2
E  electric field (volt/m)
D  electrical displacement (coulomb/m2)
  dielectric constant (coul/volt-m)
k m  mechanical (short-circuit) spring constant
m  k m / m  mechanical (short-circuit) resonant frequency
C e  A p / t p  electrical (   0 ) capacitance
Cmc  Ce /(1   )  motion constrained (   0 ) capacitance.
L  inductance
 mc  1 / LCmc  motion constrained resonant frequency
  mechanical damping factor (force/velocity)
m m
 mechanical Q-factor
Qm 

GL  1 / RL  load conductance
GL
 normalized load conductance
 m C mc
Gin  1 / Rin  internal conductance
G LN 
J. Low Power Electron. Appl. 2013, 3
GinN 
197
Gin
 Qm  normalized internal conductance
 m C mc
Refer to the device of Figure 2. When the output is shorted, E = 0, and the mechanical stiffness is
given by Young’s modulus, k m  A p Y / t p . The short-circuit, resonance frequency is given by
 m2  k m / m . However, when the output is in the open circuit condition, D = 0; the open-circuit
stiffness is given by koc  km (1   ) , where  is the dimensionless PZ coupling constant, defined in
Table 1. From this, it can be shown that the open-circuit and short-circuit resonant frequencies oc and
 m are related by the equation  oc2   m2 (1   ) . This relationship is well-known, and has been used to
experimentally determine the coupling constant ρ [1].
Similarly, we define the electrical capacitance C e  A p / t p for the case when there is no stress
(  0) ; and we define the motion constrained capacitance Cmc  Ce /(1   ) for the case when δ = 0.
The equations for the PZ EHD shown in Figure 2 are given below
F  ma  F   k m X  k m dV  j X   2 m ( X  Z )
i
(3)
 1
1 
dQ
V

  jQ   jC mcV  jk m dX  

j
L
R
dt
L 

(4)
These equations can be solved for V ( ) and X ( ) as shown below.
V 
X 
  4 Z / d
 2
j m
  m   2 
Qm


 2
  mc   2  j m G LN   2 m2



(5)

  2  j m G LN  2 Z
 2
j m  2
  m   2 
  mc   2  j m G LN   2 m2
Q
m


2
mc


(6)
When the source vibration frequency ω equals the mechanical resonant frequency  m , Equation (5)
for output voltage reduces to a familiar form.
V 
Ip
G L  Gin  j m C mc  ( j m L)
1

Qm Z / d
2
  mc

G  G  j 1  2
m

N
L
N
in




where Gin  Qm m C mc and I p  G in Z / d .
This results in the familiar circuit model for the PZ EHD, shown at the left of Figure 3.
(7)
J. Low Power Electron. Appl. 2013, 3
198
Figure 3. The circuit within the dashed box is the equivalent circuit that applies to a PZ
EHD when =m. At this frequency, the PZ current I p is independent of the load. For
 ≠ m this equivalent circuit does not accurately model device behavior, because the PZ
current I p changes as the load changes. The circuit at the right includes an inductor to
cancel the capacitive reactance of the EHD.
+
IP
Cmc
Rin
L
V
RL
‐
If a purely resistive load is connected to the EHD, the device capacitance Cmc degrades output
power. As a result, an inductor (or an effective inductor) is added to the output circuit for the purpose
of cancelling the capacitive admittance and achieving maximum average power to the load.
2
1 Z 
av
Pmax
   Gin
8 d 
(8)
In succeeding sections, simulations of voltage and output power are shown as a function of
frequency. In [2] simulations were shown for   0.2 ; Qm  50 ; and G inN  Q m  10 . Throughout
this paper, simulations are shown for   0.05 ; Qm  20 ; and G inN  Q m  1.0 . These values are
more representative of today’s commercial devices.
Figure 4. Voltage magnitude for the case of no inductor. Voltage is normalized to Z / d ,
the open-circuit voltage at w  1 .
Log10(Normalized Voltage Magnitude)
0.5
=0
0.0
  0.05 Qm  20
GinN  Gin /(Cmcm )  Qm  1
‐0.5
=0.1
‐1.0
=1
‐1.5
=10
‐2.0
‐2.5
0.7
0.8
0.9
1.0
1.1
1.2
1.3
w=ω/ωm
Figure 4 shows the magnitude of the output voltage as a function of frequency, calculated using
Equation (5), for the case of no inductor (  mc  0 ). On the vertical axis, voltage is normalized to the
J. Low Power Electron. Appl. 2013, 3
199
maximum open-circuit voltage at the mechanical resonant frequency Vocmax  Z / d . Frequency, on the
horizontal axis, is normalized to  m . For w   /  m  1 , Figure 4 confirms the predictions of
Equation (7). For GLN  0 , the normalized voltage magnitude is 0.707; and for G LN  G inN , the
normalized voltage magnitude is 0.447. However, for w  1 , Figure 4 shows interesting behavior,
especially for G LN  G inN . For GLN  0 , the voltage shows a resonance at  oc   m 1    1.025 m .
Figure 4 shows that, for large GLN , the voltage peaks at the mechanical or short-circuit resonance
frequency  m . However, for small GLN , the device characteristics change. When GLN  0 , the peak
voltage occurs at  oc   m 1    1.025 m . This can be understood as follows. For large GLN , the
electric field is effectively shorted. As GLN decreases, the electric field in the EHD increases, and alters
the effective spring constant of the cantilever beam.
It is tempting to assume from the above that frequency tuning is possible only in the narrow range
m    oc [4]. However, as we will see below, the addition of a resonant electrical circuit allows the
voltage to swing below zero and above Voc , thereby enabling a wider tuning range.
3. Coupled Oscillators
In the simulations in the previous section, we did not attempt to cancel the reactive admittance of
the PZ capacitor, and we observed the degradation in output voltage. Since G LN  G L /( m C mc )  1 in
the simulations above, the degradation is not large. However, in some cases, the PZ EHD has a large
capacitance, which can substantially degrade output power at    m . In Figure 5, we simulate
Equation (5) for the case  mc   m . This is equivalent to adding an inductor that cancels the reactive
admittance of the capacitor at    m . The inductor performs as expected at    m . The output
voltage equals Vocmax . Moreover, for large GLN , the output voltage continues to show a resonant peak at
   m ; and the voltage falls away sharply away from    m .
However, for    m , Figure 5 shows a surprising result when GLN is small. Two peaks occur at
w    /  m  1.118 and w   / m  0.894 . These resonances result from the poles in the
denominator of Equation (5) when GLN  0 .

2
m


2
  2  mc
  2   2  m2  0
(9)
Solution to the pole-splitting equation above, for the case  mc   m and   0.05 gives the values
of w and w above. Note that, when  mc  0 (case of no inductor), the solutions of Equation (9) are
w  0 and w  1    1.025 . Pole splitting, which describes coupled modes of the mechanical
and electrical resonators, occurs only for small GLN . When GLN is large, the electric field in the PZ
material is screened, and coupling is suppressed.
J. Low Power Electron. Appl. 2013, 3
200
Figure 5. Voltage magnitude for the case when an inductor of value L  ( m2 C mc ) 1 is
added to the circuit. Voltage is normalized as in Figure 4.
  0.05 Qm  20
GinN  Gin /(Cmcm )  Qm  1
Log10(Normalized Voltage Magnitude)
1.0
=0
=0.01
0.5
0.0
=0.1
‐0.5
=1
‐1.0
‐1.5
=10
‐2.0
‐2.5
0.7
0.8
0.9
1.0
1.1
1.2
1.3
w = ω/ωm
Figure 6. Roots of the Pole-splitting Equation (9). The normalized pole frequencies
w   / m and w   / m are plotted vs. wmc  mc / m .
2.0
=0.2
Normalized Pole Frequencies
w+ =ω+/ωm and w‐=ω‐/ωm
1.8
=0.05
=0.01
1.6
1.4
1.2
=0.01
1.0
=0.05
=0.2
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
wmc = ωmc/ωm
The roots of the pole-splitting equation are shown in Figure 6 for several values of coupling
constant  . In Figure 5, we selected  mc   m , in order to optimize output power at    m . But, we
J. Low Power Electron. Appl. 2013, 3
201
discovered that, if we increased the load resistance, we could also optimize output voltage at w
and w . (We will show in the next section that output power is also optimized at these frequencies).
This analysis suggests that we can tune the EHD resonant frequency by varying  mc .
We can gain further insight into the pole-splitting by returning to Equations (5) and (6). For small
G , the following relationship holds between X ( ) and V ( ) .
N
L
dV 
2
 mc
  2 X
  2 X
 2
2
  2  j mc G LN  mc
(10)
Using Equation (10) the force of the spring can be written as

 2
Fspring  k m X  k m dV  k m 1  2
2
  mc  

 X

(11)
The above equation shows that, by tuning L, we can vary  mc and vary the effective spring constant.
When    mc , V ( ) has a phase of 180° relative to X ( ) ; and the voltage reduces the effective
spring constant. When    mc , V ( ) has a phase of 0° relative to X ( ) ; and the voltage increases
the effective spring constant. When    mc , the magnitude of X ( ) is zero. If we adjust  mc such
that the corresponding root of Equation (9) equals the source vibration frequency, then Equation (11)
reduces to
Fspring
2
 k m  2
 m

 X

(12)
Equation (12) shows that the effective spring constant can be tuned over a wide range.
It is also somewhat surprising that the peak voltages at   and   are 3.5× to 5.5× higher than Vocmax
. The simulations in this paper assume that the source magnitude Z is held constant as the frequency
changes. As a result, the input acceleration increases in proportion to  2 , and V (  )  V (  ) . The
important result is that the peak voltages away from resonance can be somewhat higher than Vocmax , and
the higher voltage enables frequency tuning.
4. Optimizing Output Power
In the previous section, we showed that voltage can be made to peak at frequencies   and   ,,
which are different from m . Figure 7 shows that power is also maximized at these frequencies. The
av
curve for G LN  GinN indicates that output power peaks at    m , at the power Pmax
given by
Equation (8). The curve for G LN  10G inN shows a peak at    m at a degraded power level. The curve
av
for G LN  0.1G inN shows two peaks at   and   that have output power comparable to Pmax
. When GLN
is further reduced to G LN  0.01GinN , the output power at   and   decreases.
J. Low Power Electron. Appl. 2013, 3
202
Figure 7. Normalized average output power for the case when an inductor of value
av
Equation (8).
L  ( m2 C mc ) 1 is added to the circuit. Average power is normalized by Pmax
0.5
Log10(Normalized Average Power)
0.0
‐0.5
‐1.0
=1
=0.1
‐1.5
=10
‐2.0
  0.05 Qm  20
GinN  Gin /(Cmcm )  Qm  1
‐2.5
=0.01
‐3.0
‐3.5
‐4.0
0.7
0.8
0.9
1.0
1.1
1.2
1.3
w = ω/ωm
Figure 7 suggests that output power can be optimized at frequencies different from  m by adjusting
the external inductor (or effective inductor). Equation (5) provides a general expression for V ( ) .
Varying the parameter  mc  ( LC mc ) 1 in Equation (5) is equivalent to varying the inductor.
Maximizing power with respect to  mc gives the expression
2
 2 
 mc
( m2   2 ) 2 m2 
(   ) 
2
m
2 2
 2 m2
Q
(13)
2
m
When we use Equation (13) to determine the value of  mc that optimizes power at each frequency,
the resulting voltage and power are shown in Figures 8 and 9.
Equation (13) suggests that we need two strategies for optimizing power, depending on the source
frequency  . For frequencies near  m , (Region 2), Equation (13) reduces to
2
 mc
  2  ( m2   2 ) Q m2
(14)
Note that, when    m , Equation (14) reduces further to  mc   m , which is equivalent to
matching the capacitive admittance (refer to Figure 3). When  is above or below  m (Regions 1 and
3), Equation (13) reduces to the pole-splitting Equation (9). Far from  m , output power is maximized
at the pole frequencies [roots of Equation (9)]. However, as  approaches  m , interaction between the
poles shifts the max-power frequency [given by Equation (13)] slightly away from the pole frequency.
J. Low Power Electron. Appl. 2013, 3
203
Figure 8. Normalized voltage magnitude for the case when the reactive admittance is
optimized to give maximum output power using Equation (13). Plots of Voltage vs.
frequency are given for different values of normalized load conductance GLN . These plots
can be thought of as envelopes of curves such as Figure 5 for different values of mc .
Log10(Normalized Voltage Magnitude)
1.5
=0
  0.05 Qm  20
GinN  Gin /(Cmcm )  Qm  1
1.0
=0.01
0.5
=0.1
0.0
‐0.5
=1
‐1.0
‐1.5
=10
‐2.0
‐2.5
0.7
0.8
0.9
1.0
1.1
1.2
1.3
w = ω/ωm
Figure 9. Normalized average power for the case when the reactive admittance is
optimized to give maximum output power using Equation (13). Plots of power vs.
frequency are given for different values of normalized load conductance GLN . Power increases
with increasing ω because source acceleration increases, thereby increasing input power.
=0.01
Log10(Normalized Average PowerP)
0.5
0.0
=0.1
‐0.5
‐1.0
=1
‐1.5
‐2.0
=10
  0.05 Qm  20
GinN  Gin /(Cmcm )  Qm  1
‐2.5
Region 1
‐3.0
Region 2
Region 3
‐3.5
0.7
0.8
0.9
1.0
w = ω/ωm
1.1
1.2
1.3
J. Low Power Electron. Appl. 2013, 3
204
Within the region w   ( 2Q m ) 1 around the mechanical resonant frequency (Region 2), output
power can be optimized by using (14) for reactive admittance and G L  Gin . In Regions 1 and 3,
power is optimized by using the pole splitting Equation (9) for reactive admittance and GL  Gin . For
the parameters used in this example ( Qm  20 ), w  0.025 .
Cammarano et al. [3] have derived an equation very similar to Equation (13): Equation (8) in [3].
They observe that the power conditioning system at the output of the EHD can be used to synthesize
the complex load impedance required by Equation (13), and they comment on the challenge of
reducing the power of such systems. Chang et al. [5] have implemented a switch-mode power
conditioning system to the output of a PZ EHD, and have demonstrated the ability to harvest energy
from two sources simultaneously:    m and   1.2 m .
5. Bias-Flip Technique
For a typical, discrete EHD, C mc  100nF , and the inductor required to match this reactance at
100 Hz is impractically large: L  25 H . However, it has been shown that the Bias-Flip technique can
be used to synthesize a reactive impedance for effective impedance matching [6,7]. This technique is
suitable to ULP miniaturization. It utilizes a very small inductor together with ULP microelectronics to
emulate an inductor that is large and tunable. The BF technique has been shown to be effective in
maximizing the output power of PZ EHDs at    m [7]. In this section, we will describe the
Bias-Flip technique in the context of the equivalent circuit of Figure 3 describing a PZ EHD operating
at the mechanical resonance frequency.
Figure 10. Operation of a Bias Flip (BF) Inductor. (a) A small inductor is connected to the
output through ideal switches; (b) When the switchers are closed, the LC tank circuit begins
to oscillate, when the switches are opened, half a period later, the sign of the voltage has
been adiabatically “flipped”; (c) To achieve maximum power to the load, the bias is flipped
when Ip changes sign; (d) The resulting voltage waveform is “in-phase” with the current.
J. Low Power Electron. Appl. 2013, 3
205
The BF technique is illustrated in Figure 10. In the BF circuit, the large inductor is replaced by a
small inductor, connected by MOS switches. When the switches are closed, a high frequency tank
circuit is formed. After ½ period of oscillation of this tank circuit, the switches are opened, and the
voltage on the capacitor has “flipped” adiabatically from +V to −V. In this paper, the switches are
assumed to be ideal and lossless.
Refer to Figure 3. When an ideal inductor is used together with a matched resistive load RL  Rin ,
av
 I p2 /(8Gin ) . [See Equation (8)] In Figure 11, we
the maximum average output power is given by Pmax
show how effective the ideal Bias-Flip circuit is in achieving maximum power.
Figure 11. Normalized Average Power delivered to the load at    m . The equivalent
circuit of Fig 3 is used with RL  Rin . The dashed line shows the case of no inductor. The
solid line shows the improvement in output power that is achievable using the Bias-Flip
inductor. Both curves are normalized to the average power obtained using an ideal tuned
av
 I p2 /(8Gin ) .
inductor Pmax
Log10(Cmc)
‐8.0
‐7.5
‐7.0
‐6.5
‐6.0
‐5.5
‐5.0
Log10(Normalized Average Power)
0.0
‐0.5
m
 100 Hz
2
1
Rin  Gin  10 k
‐1.0
‐1.5
No Inductor
PNIav
4Gin2

av
Pmax
4Gin2 (mCmc)2
Bias Flip
av
PBF
 8/ 2
av
Pmax
‐2.0
‐2.5
‐3.0
In the worst case of very large C mc , the output power is degraded by several orders of magnitude,
av
 I p2 /( 2 Gin ) , which is
when no inductor is used. However, the Bias-Flip approach delivers power PBF
8 /  2  81% of the max power obtained using an ideal inductor. This illustrates the effectiveness of
Bias-Flip circuits to achieve high output power when C mc is large [7].
So far in this paper, we have discussed the case in which AC power is delivered to a resistive load.
We have done this because the analysis can be performed in closed form. However, in many energy
harvesting applications, it is necessary to rectify the AC power and store it in a battery or
super-capacitor. The Bias-Flip technique is especially applicable to this case, as shown in [7].
The rectification circuit analyzed in [7] is shown in Figure 12. For simplicity, we assumed that the
EHD is operating at the mechanical resonance frequency, and we use the equivalent circuit of Figure 3.
The output AC voltage v (t ) is rectified in the diode bridge and stored on the capacitor C RECT that is
maintained at voltage VRECT by the Energy Management Circuit. The analysis below assumes
ideal diodes.
J. Low Power Electron. Appl. 2013, 3
206
Figure 12. Circuit to rectify and store the AC power being generated by the EHD. It is
assumed that the EHD is operating at the mechanical resonance frequency.
vRECT
D1
IP
Rin
Cmc
Bias
Flip
Circuit
D2
CRECT
Energy Management Circuit
D3
D4
Operation of the Bias-Flip rectifier is described with reference to Figure 13.
Figure 13. (a) Voltage waveform v (t ) in Figure 12 for the case C mc  0 ; in this case, the
Bias-Flip circuit is not required; (b) Voltage waveform for the case of large C mc without
(a)
Voltage
Bias-Flip compensation; when the current becomes positive, there is a large negative
voltage on the capacitor C mc , this must be discharged before the voltage can swing
positive; (c) When the current becomes positive, the polarity of the voltage v (t ) is
“flipped”. This reduces the time to diode turn-on and increases power transferred to CRECT .
Rin i p (t )
VRECT
t
ton
toff
(b)
Voltage
Cmc = 0
Rin i p (t )
VRECT
t
ton
Neg
Bias
Large Cmc
No BF
toff
J. Low Power Electron. Appl. 2013, 3
207
Voltage
Figure 13. Cont.
(c)
Rin i p (t )
VRECT
t
ton
Bias
flip
toff
Large Cmc
With BF
When the capacitance is zero, as shown in Figure 13a, Bias-Flip is not required. v (t )  I p Rin sin(t )
until t  t on ; at which time, v (t ) becomes clamped at VRECT . Between t on and t off power is supplied to
the storage unit. At t  t off , the diodes turn off, and v (t ) returns to zero following the curve
v (t )  I p Rin sin( t ) . The presence of non-zero C mc degrades transferred power: Figure 13b. When the
current turns positive, there is a negative bias on C mc that must be discharged before the voltage can
swing positive. This delays diode turn-on, and forces a reduction in VRECT , both of which degrade
transferred power. This degradation can be corrected by adiabatically flipping the bias on C mc when
the current changes sign, as illustrated in Figure 13c.
Figure 14. Power transferred to the storage capacitor C RECT as a function of VRECT for
av
(see Equation (8)). The curve for
different values of C mc . Power is normalized to Pmax
Cmc  0 is negligibly different from the curve for Cmc  10nF , and is not shown. For the
case of no capacitor, the max power transfer of 0.92 occurs at V RECT  0.4 I p Rin .
1.0
Cmc=10nF
0.9
m
 100Hz
2
1
Cmc=100nF
0.8
Rin  Gin  10k
0.7
Cmc=200nF
PRECTIN
0.6
0.5
0.4
Cmc=500nF
0.3
0.2
Cmc=1uF
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
VRECT /(IpRin)
0.6
0.7
0.8
0.9
1.0
J. Low Power Electron. Appl. 2013, 3
208
The energy transferred per cycle depends on VRECT . When VRECT is low, the power transfer interval
t off  t on is long, but the power is low. When VRECT is high, the power transfer is high, but the transfer
interval is short. In fact, for VRECT above a maximum value the diodes do not turn on, and no power is
transferred. Figure 14 shows the power transfer as a function of VRECT , for various values of C mc . This
simulation is made using the values Rin  10k and  m /(2 )  100Hz .
Figure 15. Shows the rectified power as a function of C mc . For each point on the curve,
av
VRECT was selected to give the max power transfer. Power is normalized to Pmax
, and
capacitance has units Farads.
8% gap
‐8.0
‐7.5
‐7.0
‐6.5
‐6.0
‐5.5
‐5.0
0.0
Using Bias Flip
Log10 (Normalized Power) ‐0.2
‐0.4
‐0.6
‐0.8
‐1.0
m
 100Hz
2
1
Rin  Gin  10k
No Bias Flip
‐1.2
‐1.4
‐1.6
‐1.8
Log10(Capacitance )
Figure 15 shows that, for large C mc , output is severely degraded. However, the Bias-Flip circuit is
effective in recovering most of the lost output power. This analysis confirms the conclusions of
Ramadass and Chandrakasan [7] that for an EHD, operating at resonance, the Bias-Flip circuit is
effective in canceling the reactive impedance of the device, and achieving near-optimum output power.
In the next section, we will demonstrate that the Bias-Flip technique can be used to form an effective
LC tank circuit that, when coupled to the EHD can tune the resonant frequency.
6. Bias Flip for Frequency Tuning
In Section 5, we confirmed the effectiveness of the BF technique for power optimization at the
mechanical resonant frequency. When   m , and the equivalent circuit of Figure 3 applies, we can
optimize power to the load by “canceling” or “matching” the reactive admittance of the capacitor with
an inductor. We select an inductor value such that  mc  ( LC mc ) 1 / 2   m . Matching the reactive
admittance aligns the phase of the voltage across the load with the phase of the current. This works
because the current source is assumed to be ideal. Varying the reactive load does not change the
J. Low Power Electron. Appl. 2013, 3
209
current from the source. We confirmed the finding of [7] that the BF inductor is effective in canceling
capacitive admittance at  m . In this section, we will demonstrate that the BF inductor can also be used
to tune the resonant frequency of the EHD and optimize power at frequencies substantially different
from  m .
In order to maximize output power at any frequency, we need to maximize the input power
delivered from the mechanical source to the EHD. In other words, we need to align the phase of the
force with the phase of the source velocity.
In the following analysis, we assume the phase of z (t ) to be zero. z (t )  Z cos(t ) , and the
velocity of the source is dz / dt  Z sin(t ) . The source velocity has a phase of +90o. The force
acting on the EHD is given by Equation (3). Our goal is to maximize.
Pi av 
1

Fdz  FI Z

2
2
(15)
Where FI is the imaginary part of F . F (t )  FR cos(t )  FI sin(t ) . From this, we conclude that
input average power is maximized when F has phase +90°, matched to source velocity, and X has
phase −90°.
Figure 16. Phase of mechanical displacement X ( ) for three cases: (1) No inductor;
(2) Inductor optimized using (13) and (3) Inductor optimized using the pole-splitting
Equation (9). In all three cases, GL  0 .
Phase of Mechanical Displacement X (deg)
0
‐20
‐40
‐60
Optimized Inductor
‐80
‐100
Pole‐Splitting Equation
‐120
‐140
‐160
No inductor
‐180
‐200
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
w = ω/ωm
Additional insight into maximization of output power is seen in Figure 16. Here, we compare the
phase of mechanical displacement X ( ) for three conditions
1. No inductor. The phase is −90° only at   oc   m 1   . Below  m , the phase is ≈0°, and
above  m , the phase is ≈ −180°. Only at   oc is the force in phase with the source velocity;
2. Inductor, optimized using Equation (13). Note that the phase approaches −90° above and below
  m ;
J. Low Power Electron. Appl. 2013, 3
210
3. Inductor optimized using the pole-splitting Equation (9). The phase is −90° for all frequencies.
The improvement in power at frequencies above and below  m results from phase alignment
between force and source velocity.
Figure 17 shows output power for 3 conditions.
1. No inductor: G LN  G inN  1 ;
2. Inductor optimized using Equation (13): G LN  G inN  1 ;
3. Inductor optimized using Equation (13): GLN  0.1 .
Figure 17. Normalized Average Output Power for three cases. (1) No inductor &
G LN  G inN  1 ; (2) Inductor optimized using Equation (13) & G LN  G inN  1 ; (3) Inductor
optimized using Equation (13) & GLN  0.1 .
0.5
Log10(Normalized Average Power)
Optimized Inductor
=0.1
0.0
‐0.5
Optimized Inductor
=1
‐1.0
‐1.5
=1
No inductor
‐2.0
‐2.5
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
w = ω/ωm
Case #2 illustrates the case where the reactive admittance is chosen to optimize output power, but
the load conductance GLN is kept at G LN  G inN  1 . Very little improvement is achieved, because the
voltage is kept low by the high load conductance, and the voltage is ineffective in modulating the
cantilever spring constant. Case #3 shows power improvement of ~50X compared to case #1. When
we compensate for the increase in acceleration with frequency, case #3 demonstrates that it is possible
to achieve output power at    m that is comparable to the maximum power at  m .
Additional insight into the mechanism of frequency tuning can be obtained by transforming the
mechanical equations of motion to an equivalent circuit [1,8,9], as shown in Figure 18.
J. Low Power Electron. Appl. 2013, 3
211
Figure 18. (a) Equivalent circuit model for a PZ EHD in which Lm  m , Rm   ,
2
C m  k m1 , VF  m Z , I s ( )  S ( )  jX ( ) , and A  k m d The subscript m
denotes the mechanical circuit. S ( ) is used for velocity to avoid confusion with voltage;
(b) Equivalent circuit model of Figure 18a, in which the electronic circuit is replaced by an
electrical impedance Ze.
Cm
Lm
+
Av(t)
is(t)
+
‐
Rm
Ais(t)
+
Cmc
vF(t)=VFcost
RL V(t)
L
‐
‐
(a)
Cm
Lm
+
Rm
Is(ω)
VF()
Ze 
A2Ze
‐
V(ω)
Cmc
L
RL
(b)
The equations for the mechanical portion of the equivalent circuit are shown below.


k


1
VF   jLm 
 Rm  A 2  e ( )  I s  m 2 Z   jm  m   ( jX )  k m dV
j
jC m




(16)
where  e , defined in Figure 18, is given by
e 
jCmc
1
j L

2
2
 1 /( jL )  1 / RL 1   / mc  jL / RL
(17)
Define Z m to be the impedance seen by the voltage source VF ( ) in Figure 18b. Setting
Im( m )  0 gives the pole-splitting equation, equivalent to Equation (9), in the limit RL  L .


2
1
A2 L 1   2 / mc
Im(m )  Lm 

0
2
2 2
Cm 1   2 / mc
 L / RL 


(18)
The last term in the above equation can be used to tune the resonance frequency above or below the
mechanical resonance frequency. The last term takes the form
2
jA2 L(1   2 / mc
)
 jLeff
2
2 2
(1   / mc )  (L / RL ) 2
(19)
J. Low Power Electron. Appl. 2013, 3
212
where Leff can be made positive or negative to add or subtract from Lm . At the mechanical resonance
frequency    m , Im(  m )  Lm 
1
 0 , and output power is optimized by setting    mc .
C m
When    m the resonant frequency   must be reduced. This can be achieved by increasing the
effective inductance, which can be achieved by setting  mc   . When    m , the resonant
frequency   must be increased. This can be achieved by decreasing the effective inductance, which
can be achieved by setting mc   .
These results are summarized in Table 1.
Table 1. Strategy For Power Optimization in the Three Frequency Regions of Operation.
Frequency region
Region 1: ω < ωm
Region 2: ω ≈ ωm
Region 3: ω > ωm
Leff
>0
≈0
<0
ωmc
ωmc > ω
ωmc = ω ≈ ωm
ωmc < ω
Phase of voltage
~ +90o
~0o
~ −90o
Optimum power
RL >> Rin
RL = Rin
RL >> Rin
Maximizing power in the three regions can be envisioned in term of an effective inductor.
Alternatively, it can be envisioned in terms of setting the phase of the voltage V ( ) . The phase in each
of the three regions is given in the table. In Region 2, the voltage across the load V ( ) is in phase with
I s ( ) , and output power is optimized by setting R L  Rin . However, in Regions 1 and 3, max power
occurs when V ( ) has a phase of +90o (Region 1) and −90° (Region 3) relative to the phase of I s ( ) ,
and optimum power is achieved by setting RL to be substantially larger than Rin .
The foregoing analysis suggests that the Bias-Flip technique can be used to synthesize an inductor,
by flipping the polarity of the voltage in such a way that X ( ) has a phase of −90° in all three regions
of operation. Recall from Equation (10) that, when GLN is small, the phase of X ( ) is related to the
phase of V ( ) in a simple way. By adjusting the phase of V ( ) as shown in Table 1, we also adjust the
phase of X ( ) to −90°.
Simulations were performed starting from the differential equations for v (t ) and x(t ) that are
comparable to Equations (3) and (4), for the case of no inductor. These equations were solved subject
to the boundary conditions.
v(t  T / 2)  v(t  0) x(t  T / 2)   x(t  0) and
dx
dx
(t  T / 2)   (t  0)
dt
dt
(20)
In the case of no Bias-Flip, v(t  T / 2)  v(t  0) . The effect of the BF inductor is to change the
phase of v(t) every half-period.
av
Using the voltage waveform, we calculated average output power. This is shown normalized to Pmax
in Figure 19. These simulations show that the BF technique is capable of achieving output power,
comparable to the optimum power achievable with an optimized inductor. Moreover, the BF technique
is self-tuning. If the bias is flipped whenever the source velocity crosses zero (as assumed in this
simulation), the desired phase is maintained as the source frequency changes. No calculation is
required to solve Equation (13).
J. Low Power Electron. Appl. 2013, 3
213
Figure 19. Normalized average power as a function of frequency. In Regions 1 and 3, the
Bias-Flip technique (red and blue dashed lines) improves output power by ~100X
compared to the case of no inductor and no BF (green line). Moreover, it gives output
power that is comparable to the maximum power achievable with an optimized inductor
(red and blue solid lines).
1.0
=0.01
Log10(Normalized Average Power)
0.5
=0.01
0.0
=0.1
‐0.5
Bias‐Flip:
‐1.0
=0.1
‐1.5
‐2.0
Region 2
‐2.5
‐3.0
Region 1
Region 3
‐3.5
‐4.0
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
w = ω/ωm
Analysis of the voltage waveforms reveals another aspect of self-tuning. In Region 1, the bias flips
from negative to positive at t  0 and from positive to negative at t  T / 2 , thereby emulating a +90°
phase shift. In Region 3, the reverse happens. The bias flips from positive to negative at t  0 and
from negative to positive at t  T / 2 , thus emulating a −90° phase shift.
7. Conclusions
In the preceding sections, we have explained the principles for electrically tuning of PZ EHDs.
These principles are summarized below.
Equation (11) shows that the effective spring constant of the mechanical resonator is a function of
voltage. If the load conductance is large, the voltage is kept small, and the resonator responds only at
the mechanical resonant frequency  m . However, for small load conductance G L , the voltage can be
used to tune the spring constant, and the resonant frequency of the mechanical oscillator.
In Regions 1 and 3, output power is maximized by maximizing input power (force x velocity),
transferred from the source to the EHD. At frequencies below  m (Region 1), this occurs when the
phase of the voltage is +90° relative to the source vibration, and at frequencies above  m (Region 3),
output power is optimized when the phase of the voltage is -90o relative to the source vibration.
This optimum phase relationship can, in theory, be achieved using a tunable inductor, whose value
can be obtained from Equation (13). A large tunable inductor is not generally practical. However, the
Bias-Flip technique can be used to emulate a large, tunable inductor. Previous work has shown that the
J. Low Power Electron. Appl. 2013, 3
214
BF technique can be used to optimize the output power at  m , by cancelling the capacitive admittance
of the EHD. In this work, we have shown how the BF technique can be used to tune an EHD and
harvest energy from frequencies other than the mechanical resonance frequency.
Acknowledgments
The authors acknowledge informative exchanges with Samuel Chang of MIT.
Conflict of Interest
The authors declare no conflict of interest.
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© 2013 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article
distributed under the terms and conditions of the Creative Commons Attribution license
(http://creativecommons.org/licenses/by/3.0/).